The Ramanujan Library -- Automated Discovery on the Hypergraph of Integer Relations

TL;DR

The paper targets automatic discovery of interrelations among fundamental constants by introducing a hypergraph framework where vertices are constants and edges encode polynomial relations via $C$-transforms for generalized continued fractions. Automated enrichment relies on the PSLQ algorithm, guided by a Return on Investment (RoI) metric to filter significant results, and employs an identify utility to grow the database, all within a public open-source Ramanujan Library. The key contributions include the first public database of constants and their interrelations, a canonical $C$-transform representation to capture broad formula classes, and the discovery of 75 novel connections (alongside Ramanujan-like families) demonstrated through a scalable, parallelizable workflow. This resource enables reproducible, cross-disciplinary exploration of constants and provides a platform for algorithm development in experimental mathematics. The work illustrates how a knowledge-graph-like structure can organize deep mathematical relations and accelerate conjecture generation beyond traditional, human-centered approaches.

Abstract

Fundamental mathematical constants appear in nearly every field of science, from physics to biology. Formulas that connect different constants often bring great insight by hinting at connections between previously disparate fields. Discoveries of such relations, however, have remained scarce events, relying on sporadic strokes of creativity by human mathematicians. Recent developments of algorithms for automated conjecture generation have accelerated the discovery of formulas for specific constants. Yet, the discovery of connections between constants has not been addressed. In this paper, we present the first library dedicated to mathematical constants and their interrelations. This library can serve as a central repository of knowledge for scientists from different areas, and as a collaborative platform for development of new algorithms. The library is based on a new representation that we propose for organizing the formulas of mathematical constants: a hypergraph, with each node representing a constant and each edge representing a formula. Using this representation, we propose and demonstrate a systematic approach for automatically enriching this library using PSLQ, an integer relation algorithm based on QR decomposition and lattice construction. During its development and testing, our strategy led to the discovery of 75 previously unknown connections between constants, including a new formula for the `first continued fraction' constant $C_1$, novel formulas for natural logarithms, and new formulas connecting $π$ and $e$. The latter formulas generalize a century-old relation between $π$ and $e$ by Ramanujan, which until now was considered a singular formula and is now found to be part of a broader mathematical structure. The code supporting this library is a public, open-source API that can serve researchers in experimental mathematics and other fields of science.

Figures (7)

• Figure 1: Automatically discovered formulas, showing a section of the full hypergraph in figure \ref{['fig:hypergraph']}. The first formula was found by Ramanujan berndt_ramanujan_problems, and the second formula is novel.
• Figure 2: Automated search of integer relations. The process begins by collecting fundamental constants and continued fractions from the literature and organizing them into a database. Then, the algorithm checks subsets of constants for polynomial relations using PSLQ. We identify that a relation is significant by a high Return on Investment (RoI), as described in section \ref{['pslq-roi']}. Such events are generally extremely rare, and each one is returned to the database and saved, enriching the hypergraph and adding to our knowledge about mathematical constants and their relations. Each such discovered relation also saves time on future runs of PSLQ. In addition, our novel identify utility (see section \ref{['api']}) allows for manually adding constants and continued fractions to the database regardless of the automated search.
• Figure 3: Experimental analysis of the Return on Investment (RoI) property, showing its use for identifying integer relations. (a) For each $n$, we ran PSLQ $100$ times with a pre-selected binary precision of $50+5n$. We present the average RoI for each $n$. The standard deviation for each $n$ is presented as fading errorbars, with the half length of each darkest error bar being equal to one standard deviation. The lighter dashed line is the constant RoI of $1.25$ and the darker dashed line is the constant RoI of $1.5$, which the plot suggests are viable options for minimum RoI for filtering integer relations. (b) Sampled formulas are listed with their RoI on panel (a). Thanks to their high precision, their RoI is much greater than our recommended RoI cutoff. (c) For each $d,n$, we ran PSLQ $100$ times until tolerance (equal to $65\%$ of the working precision, see appendix \ref{['pslq-rebounding']} for more details), each with $n$ numbers between $0$ and $1$, each with $d$ uniformly random binary digits. Then, we present the average RoI across all $100$ runs for each $d,n$. For a fixed $n$, the average RoI is close to constant in $d$.
• Figure 4: The hypergraph of integer relations: each vertex is a constant, and each edge is a formula. The hypergraph summarizes our automated discovery of relations between mathematical constants, presenting selected formulas from our database. This hypergraph does not show all discovered relations for clarity's sake. Empty circles denote constants (written inside). Colored circles denote continued fractions. Black denotes continued fractions whose limit is known in the broader literature. Green denotes continued fractions whose limit is found in rm1. Blue denotes continued fractions whose limit was unknown before our work, to the best of our knowledge. We denote order-1 connections (Mobius-like) with solid lines, and higher-order connections (constants may appear squared, cubed, etc.) with dashed lines. Edges that connect more than two vertices are marked with small empty squares, denoting formulas that involve more than two constants of formulas (example in the inset at the top right). Each algebraic constant has an edge of size 1, also marked with a small empty square (placed at the top left). Such degenerate edges correspond to the minimal polynomial of the constant.
• Figure 5: Equivalences between continued fractions. Expanding on figure \ref{['fig:excerpt']}, our representation of the continued fractions enables each integer relation to capture an infinite family of formulas. This redundancy is mostly eliminated by using the $\mathcal{C}$-transform. Additional equivalences still exist, as discussed in section \ref{['api']} and shown by the second integer relation.

Theorems & Definitions (2)

• Conjecture 1
• Conjecture 2